Finite Group Databasebeta

← #10950876

Subgroups — #10950876

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81934C211yesmaximal, characteristic, complemented
2#81907C211yesmaximal, characteristic
3C3×#2255C211yesmaximal, characteristic, complemented
4#71383C211yesmaximal, characteristic, complemented
5#71353C211yesmaximal, characteristic, complemented
6#69347C212yesmaximal
7#69338C212yesmaximal, complemented
8#69328C212yesmaximal, complemented
9#69319C212yesmaximal, complemented
10#69084C211yesmaximal, characteristic, complemented
11D3×#346C211yesmaximal, characteristic, complemented
12#33747—31nomaximal, Sylow
13#5728—21no—
14#5691—21no—
15C3×#500C2211yescharacteristic
16C3×#499C2211yescharacteristic
17#5093—21no—
18#5068—21no—
19C2×Dic24C2212yes—
20C2×#800C2212yes—
21C2×D3×C16C2212yes—
22#4887—21no—
23#4887C2212yes—
24#4887C2211yescharacteristic
25#4883C2211yescharacteristic
26#4873—21no—
27#4873C2212yes—
28#4873C2211yescharacteristic
29#4872C2211yescharacteristic
30#4864—21no—
31#4852—21no—
32#4699C2211yescharacteristic
33#4696C2211yescharacteristic
34C4×D3×C8C2211yescharacteristic
35C3×#367C2212yes—
36C3×#366C2212yes—
37C3×#346C2211yescharacteristic
38#4502C2212yes—
39#4492C2212yes—
40#4492C2212yes—
41#4492C2212yes—
42#4491C2212yes—
43#4461C2211yescharacteristic
44#3170—31no—
45#3168—31no—
46#2255D311yescharacteristic, complemented
47#2251—31no—
48#2245—31no—
49#2243—31no—
50#2217—32no—
51#2216—32no—
52#2176—32no—
53#2175—32no—
54C2×#346—31no—
55C3×C4⋊Q8—21no—
56C3×#125—21no—
57#946—41no—
58Q8×Dic3—41no—
59C2×Dic12—21no—
60C2×Dic12C2312yes—
61C2×Dic12C2311yescharacteristic
62C2×D24C2311yescharacteristic
63C2×#204—21no—
64C2×D3×C8D411yescharacteristic
65C2×D3×C8C2311yescharacteristic
66#898—41no—
67#897—21no—
68#897—22no—
69#895—21no—
70#894—41no—
71#893—21no—
72#893—41no—
73#892—41no—
74#892—41no—
75#890—41no—
76#887—22no—
77#887—21no—
78#875—21no—
79C42×D3D411yescharacteristic
80#869—21no—
81C2×C3×C16C2312yes—
82C3×#107—21no—
83C3×#107C2311yescharacteristic
84C3×#107C2312yes—
85C3×#107C2311yescharacteristic
86C3×#106—21no—
87C3×C4×C8C2311yescharacteristic
88C8×Dic3D411yescharacteristic
89C8×Dic3C2311yescharacteristic
90C2×C3⋊C16C2312yes—
91#810—21no—
92#810—22no—
93#810—22no—
94#810—21no—
95#809—21no—
96#809—21no—
97C4×C3⋊C8D411yescharacteristic
98Dic24—24no—
99#800—24no—
100D3×C16—24no—
101#533—61no—
102#529—61no—
103C2×Q32—32no—
104C2×SD32—32no—
105C22×C16—32no—
106#500C2×D311yescharacteristic
107#499C2×D311yescharacteristic
108#495—31no—
109#494—31no—
110#488—61no—
111#487—61no—
112#467—31no—
113#467—32no—
114#467—61no—
115#466—31no—
116#465—61no—
117C4×Q16—31no—
118C4×Q16—32no—
119C4×Q16—61no—
120C4×SD16—61no—
121C4×D8—31no—
122C2×C4×C8—31no—
123#367C2×D312yes—
124#366C2×D312yes—
125#358—32no—
126#358—32no—
127#358—32no—
128#357—32no—
129#346—32no—
130#346—31no—
131#346C2×D311yescharacteristic
132C2×C3×Q8—41no—
133C2×D12—21no—
134C2×D12—41no—
135C2×C4×D3—41no—
136C2×C4×D3D811yescharacteristic
137C2×C4×D3D811yescharacteristic
138C2×C4×D3C2×D411yescharacteristic
139C2×Dic6—22no—
140C2×Dic6—21no—
141C2×C3×C8C2×D411yescharacteristic
142C2×C3×C8C2411yescharacteristic
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C42C2×D411yescharacteristic
150(C2×D3)⋊C4—42no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158C4⋊Dic3—42no—
159C4⋊Dic3—41no—
160Dic3⋊C4—42no—
161Dic3⋊C4—41no—
162C4×Dic3—41no—
163C4×Dic3—42no—
164C4×Dic3D811yescharacteristic
165C4×Dic3D811yescharacteristic
166C4×Dic3C2×D411yescharacteristic
167C2×C3⋊C8C2×D411yescharacteristic
168C2×C3⋊C8C2×D411yescharacteristic
169Dic12—22no—
170Dic12—22no—
171Dic12—22no—
172Dic12—42no—
173D24—22no—
174#204—42no—
175D3×C8—22no—
176D3×C8—22no—
177C3×C16#4114yes—
178C3⋊C16—22no—
179C2×Q16—32no—
180C2×Q16—31no—
181C2×Q16—61no—
182C2×SD16—61no—
183C2×D8—31no—
184C22×C8—31no—
185C22×C8—31no—
186C4⋊Q8—21no—
187C4⋊Q8—61no—
188#126—121no—
189#125—21no—
190#125—121no—
191#124—61no—
192#123—121no—
193C22⋊Q8—61no—
194C22⋊Q8—62no—
195C22⋊Q8—121no—
196#121—61no—
197C4×Q8—61no—
198C4×Q8—62no—
199C4×Q8—121no—
200C4×D4—61no—
201C4×D4—121no—
202#117—121no—
203#117—121no—
204C2×C42—31no—
205Q32—64no—
206SD32—64no—
207C2×C16—32no—
208C2×C16—64no—
209C2×C16C22×D312yes—
210#107—21no—
211#107C22×D312yes—
212#107C22×D311yescharacteristic
213#107C22×D311yescharacteristic
214#106—21no—
215#103—61no—
216#103—61no—
217#103—62no—
218#103—62no—
219#102—61no—
220#102—61no—
221C4×C8—31no—
222C4×C8—31no—
223C4×C8—31no—
224C4×C8C22×D311yescharacteristic
225C22×D3—41no—
226C22×D3C2×D811yescharacteristic
227C3×Q8—82no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4C2×D811yescharacteristic
232C2×C3×C4C2×D811yescharacteristic
233C2×C3×C4C22×D411yescharacteristic
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3C2×D811yescharacteristic
237C2×Dic3C2×D811yescharacteristic
238C2×Dic3C2×D811yescharacteristic
239D12—41no—
240D12—41no—
241D12—82no—
242C4×D3—22no—
243C4×D3—21no—
244C4×D3—21no—
245C4×D3—41no—
246C4×D3—82no—
247Dic6—42no—
248Dic6—41no—
249Dic6—41no—
250Dic6—42no—
251C3×C8#13512yes—
252C3×C8C2×#4112yes—
253C3⋊C8—21no—
254C3⋊C8—21no—
255C2×Q8—41no—
256C2×Q8—61no—
257C2×Q8—62no—
258C2×D4—61no—
259C2×D4—121no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—121no—
264Q16—62no—
265Q16—62no—
266Q16—62no—
267Q16—122no—
268SD16—122no—
269D8—62no—
270C2×C8—31no—
271C2×C8—31no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8D3×D411yescharacteristic
275C2×C8C23×D311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—121no—
283C4⋊C4—122no—
284C4⋊C4—121no—
285C4⋊C4—122no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—122no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42—31no—
295C42—31no—
296C42—31no—
297C42—121no—
298C42—122no—
299C42D3×D411yescharacteristic
300C16—62no—
301C16#23914yes—
302C22×C3C22×D811yescharacteristic
303C2×D3—21no—
304C2×D3—21no—
305C2×D3—81no—
306C2×D3—81no—
307C2×D3—81no—
308C2×D3#51012yes—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—81no—
312C3×C4—81no—
313C3×C4—82no—
314C3×C4#53211yescharacteristic
315C3×C4C2×#13511yescharacteristic
316Dic3—21no—
317Dic3—21no—
318Dic3—21no—
319Dic3—81no—
320Dic3—82no—
321C23—31no—
322C23—121no—
323Q8—82no—
324Q8—122no—
325Q8—121no—
326Q8—121no—
327Q8—122no—
328D4—121no—
329D4—121no—
330D4—242no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—62no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—122no—
343C2×C4—242no—
344C2×C4D3×D811yescharacteristic
345C2×C4D3×D811yescharacteristic
346C2×C4C2×D3×D411yescharacteristic
347C8—61no—
348C8—61no—
349C8#91912yes—
350C8C2×#23912yes—
351C2×C3#316911yescharacteristic
352C2×C3C2×#51011yescharacteristic
353C2×C3#341411yescharacteristic
354D3—22no—
355D3—161no—
356C22—32no—
357C22—61no—
358C22—61no—
359C22—241no—
360C22—241no—
361C22—241no—
362C22C2×D3×D811yescharacteristic
363C4—21no—
364C4—21no—
365C4—61no—
366C4—61no—
367C4—61no—
368C4—81no—
369C4—82no—
370C4—81no—
371C4—241no—
372C4—242no—
373C4#572511yescharacteristic
374C4C2×#91911yescharacteristic
375C3#3374711yescharacteristic, complemented, Sylow
376C2—62no—
377C2—481no—
378C2#8193311yescharacteristic
379C2#8203511yescharacteristic
380C2D3×#51011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2255C2
socleC22×C3C22×D8
prime radical#4883C22
hypercenter#2255D3
hypocenterC3#33747
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #81934C2semidirect
3. C3×#2255C2semidirect
4. #71383C2semidirect
5. #71353C2semidirect
7. #69338C2semidirect
8. #69328C2semidirect
9. #69319C2semidirect
10. #69084C2semidirect
11. D3×#346C2semidirect
46. #2255D3semidirect
375. C3#33747semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#819342
#819072
C3×#22552
#713832
#713532
#693472
#693382
#693282
#693192
#690842
D3×#3462
#337473